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Many of today’s problems in engineering demand reliable and accurate prediction of failure mechanisms of mechanical structures. Herein it is necessary to take into account the often heterogeneous structure on the fine scale, to capture the underlying physical phenomena. Despite ever increasing Computational resources, dissolving the fine scales in a direct numerical simulation is prohibitive. This work aims to develop an efficient approach to modeling nonlinear heterogeneous structures using the variational multiscale method (VMM) and model order reduction (MOR).
The VMM, introduced in, assumes an additive split of the solution into coarse and fine scale contributions. In, the VMM is applied to a damage mechanics–based material model for concrete-like materials. Herein, suitable boundary conditions for the fine scale which enable localization phenomena to evolve are discussed. As such, zero jump conditions between fine scale solutions are proposed which are enforced pointwise by a Lagrange type method leading to a coupled solution procedure.
In this contribution, possible extensions of the VMM with reduced order modeling are presented. In the linear case, assuming the fine scale solution to be zero on coarse scale element boundaries allows for static condensation and a decoupled solution procedure. Based on this, an efficient localized Training strategy will be developed. For the nonlinear case, the situation of coupled non-conforming spaces, i. e. finite element and reduced order spaces for the fine scales, arises. Thus the imposition of suitable fine scale interface conditions in the weak sense by the use of Lagrange multipliers is investigated. Specific problems in solid mechanics are used to illustrate the performance of the above approaches.
The authors gratefully acknowledge financial support by the German Research Foundation (DFG), Project number 394350870, and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC Grant agreement No. 818473).
Multiscale modeling of heterogeneous structures based on a localized model order reduction approach
(2023)
Many of today’s problems in engineering demand reliable and accurate prediction of failure mechanisms of mechanical structures. Thus, it is necessary to take into account the heterogeneous structure on the smaller scale, to capture the underlying physical phenomena. However, this poses a great challenge to the numerical solution since the computational cost is significantly increased by resolving the smaller scale in the model. Moreover, in applications where scale separation as the basis of classical homogenization schemes does not hold, the influence of the smaller scale on the larger scale has to be modelled directly. This work aims to develop an efficient concurrent methodology to model heterogeneous structures combining the variational multiscale method (VMM) [1] and model order reduction techniques (e. g. [2]). First, the influence of the smaller scale on the larger scale can be taken into account following the additive split of the displacement field as in the VMM. Here, also a decomposition of the global domain into subdomains, each containing a fine grid discretization of the smaller scale, is introduced. Second, local reduced approximation spaces for the smaller scale solution are constructed by exploring possible solutions for each subdomain based on the concept of oversampling [3]. The associated transfer operator is approximated by random sampling [4]. Herein, we propose to incorporate the actual physical behaviour of the structure of interest in the training data by drawing random samples from a multivariate normal distribution with the solution of a reduced global problem as mean. The local reduced spaces are designed such that local contributions of each subdomain can be coupled in a conforming way. Thus, the resulting global system is sparse and reduced in size compared to the direct numerical simulation, leading to a faster solution of the problem.